Ultra-compact X-ray binaries (UCXBs) are low-mass X-ray binaries with hydrogen-deficient mass-donors and ultra-short orbital periods. They have been suggested to be the potential Laser Interferometer Space Antenna (LISA) sources in the low-frequency region. Several channels for the formation of UCXBs have been proposed so far. In this article, we carried out a systematic study on the He star donor channel, in which a neutron star (NS) accretes matter from a He main-sequence (MS) star through Roche-lobe overflow, where the mass-transfer is driven by the gravitational wave radiation. Firstly, we followed the long-term evolution of the NS+He MS binaries by employing the stellar evolution code Modules for Experiments in Stellar Astrophysics, and thereby obtained the initial parameter spaces for the production of UCXBs. We then used these results to perform a detailed binary population synthesis approach to obtain the Galactic rates of UCXBs through this channel. We estimate the Galactic rates of UCXBs appearing as LISA sources to be $\sim 3.1-11.9\, \rm Myr^{-1}$ through this channel, and the number of such UCXB-LISA sources in the Galaxy can reach about 1 − 26 calibrated by observations. The present work indicates that the He star donor channel may contribute significantly to the Galactic UCXB formation rate. We found that the evolutionary tracks of UCXBs through this channel can account for the location of the five transient sources with relatively long orbital periods quite well. We also found that such UCXBs can be identified by their locations in the mass-transfer rate versus the orbital period diagram.

Although multidimensional simulations have investigated the processes of double white dwarf (WD) mergers, postmerger evolution only focused on the carbon–oxygen (CO) or helium (He) WD merger remnants. In this work, we investigate for the first time the evolution of the remnants stemming from the merger of oxygen–neon (ONe) WDs with CO WDs. Our simulation results indicate that the merger remnants can evolve to hydrogen- and helium-deficient giants with a maximum radius of about 300 R ⊙. Our models show evidence that merger remnants more massive than 1.95 M ⊙ can ignite Ne before significant mass loss ensues, and they thus would become electron-capture supernovae. However, remnants with initial masses less than 1.90 M ⊙ will experience further core contraction and longer evolutionary time before reaching the conditions for Ne burning. Therefore, their fates are more dependent on mass-loss rates due to stellar winds and thus more uncertain. Relatively high mass-loss rates would cause such remnants to end their lives as ONe WDs. Our evolutionary models can naturally explain the observational properties of the double WD merger remnant IRAS 00500+6713 (J005311). As previously suggested in the literature, we propose and justify that J005311 may be the remnant from the coalescence of an ONe WD and a CO WD. We deduce that the final outcome of J005311 would be a massive ONe WD rather than a supernova explosion. Our investigations may be able to provide possible constraints on the wind mass-loss properties of the giants that have CO-dominant envelopes.